My title - Departamento de Matemática da Universidade do Minho

10 STATISTICAL DESCRIPTION OF ORBITS 69
for any ϕ ∈ C 0 b (X, R). The invariance property ϕ (x) = (ϕ ◦ f) (x) then implies that ∫ X (ϕ ◦ f) dµ x =
∫
X ϕdµ x for any ϕ, hence that µ x is an invariant probability measure. In the language of physicists,
this says that ”time averages” along the orbit of x are equal to ”space averages” with respect to
the measure µ x .
One is thus lead to consider the following questions. Do there exist points x for which time
averages exists Given an invariant measure µ, do there exist, and how many, points x such that
µ = µ x
Example: periodic orbits. Let p be a periodic point with period n. The time average ϕ (p)
of any observable ϕ exists, and is equal to the arithmetic mean of ϕ along the orbit, namely
ϕ (p) = 1 n−1
∑
ϕ (f n (p))
n
k=0
If µ p denotes the normalized sum 1 ∑ n−1
n k=0 δ f n (p) of Dirac masses placed on the orbit of p, this
amount to say that ϕ (p) = ∫ X ϕdµ p.
Let p be a fixed point, and ϕ : X → R be an observable which is continuous at p. If x ∈ W s (p),
then the time average ϕ (x) exists and is equal to ϕ (p), i.e. time averages of points in the basin of
attraction of p are described by the Dirac measure µ p = δ p .
The Birkhoff-Khinchin ergodic theorem. Ergodic theorems are the milestones of ergodic
theory, and deal with various type of convergence of the time means ϕ n for certain classes of
observables ϕ. In particular, the Birkhoff-Khinchin ergodic theorem must be thougth as the generalization
of the Kolmogorov strong law of large numbers, as it says that time means of certain
well-behaved observables exist almost everywhere. The Birkhoff-Khinchin ergodic theorem was
actually preceeded by the von Neumann’s ”statistic” ergodic theorem, which says that
von Neumann “statistic” ergodic theorem. Let U be a unitary operator on a Hilbert space
H, let H U = {v ∈ H s.t. Uv = v} denote the closed subspace of those vectors which are fixed by
U, and P U : H → H U denote the orthogonal projection onto H U . Then, for any vector v ∈ H we
have
1
n∑
lim
U k v − P
n→∞ ∥
U v∥
= 0
n + 1
k=0
If f : X → X is an endomorphism of the probability space (X, E, µ), one can consider the
“shift” operator U : L 2 (µ) → L 2 (µ) given by (Uϕ) (x) = ϕ (f (x)). It is clearly unitary, its
fixed point set is the space of invariant L 2 -observable. The von Neumann theorem then asserts
convergence of time means ϕ n → ϕ in L 2 (µ). Here, we prove the
∥
H
Birkhoff-Khinchin “individual” ergodic theorem. Let f : X → X be an endomorphism
of the probability space (X, E, µ), and let ϕ ∈ L 1 (µ) be an integrable observable. Then the limit
1
ϕ(x) = lim
n→∞ n + 1
n∑
ϕ ( f k (x) )
exists for µ-almost any x ∈ X. Moreover, the observable ϕ is in L 1 (µ), is invariant, and satisfies
∫ ∫
ϕdµ = ϕdµ
k=0